(see [1]}). By the maximum principle, \(\,{\rm log}\,\Vert \Omega \Vert ^2\)  achieves minimum at \(t=0\) , hence it bounded below.
On the other hand, by the assumption on \(Rm\)  and \(T\) , there is constant \(C_2\) , such that the following inequality holds:
\(\partial _t \,{\rm log}\,\Vert \Omega |\Vert ^2 = \tilde{R} + {1 \over 2}|T|^2_g \le C_2.\) 
